日: 2012年12月21日

t-test on independent groups with unequal variance

Sample average from population which follows normal distribution also follows it. When standard deviation of population is not known, you would have to speculate it with standard deviation of sample. T-statistics follows t-distribution, not normal distribution.

\displaystyle t = \frac{\bar X - \mu}{SD/\sqrt n} = \frac{\bar X - \mu}{SE} \vspace{0.2in}\\

SD: standard deviation; SE: standard error

When you would like to compare average values between separate groups with unequal variance, you could calculate t-statistics with formula below;

\displaystyle t = \frac{(\bar X_2 - \bar X_1)}{SD_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}} = \frac{(\bar X_2 - \bar X_1)}{SE}\vspace{0.2in}\\ SD_p = \sqrt{\frac{(n_1 - 1)SD_1^2 + (n_2 -1)SD_2^2}{n_1 + n_2 - 2}}

SD_p; pooled SD

When t-statistics is greater than a value, null hypothesis is rejected. In one sided test, when it is greater than the value which area under t-distribution curve is smaller than 0.05, it is statistically significant. In two sided test, when it is greater than the value which area under curve is smaller than 0.025, it is statistically significant. T-statistics follows degree of freedom.

Reference:t-distribution

異分散の独立した2群のt検定

　正規分布に従う母集団からの標本平均値の分布は正規分布に従いますが，母集団の標準偏差\sigmaが未知の場合，サンプルの標準偏差から推測する必要があります．その場合，t 統計値は正規分布ではなく t 分布に従います．

\displaystyle t = \frac{\bar X - \mu}{SD/\sqrt n} = \frac{\bar X - \mu}{SE} \vspace{0.2in}\\

　異分散の独立した2群の平均値を比べる場合の t 統計値は次式で求まります．

\displaystyle t = \frac{(\bar X_2 - \bar X_1)}{SD_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}} = \frac{(\bar X_2 - \bar X_1)}{SE}\vspace{0.2in}\\ SD_p = \sqrt{\frac{(n_1 - 1)SD_1^2 + (n_2 -1)SD_2^2}{n_1 + n_2 - 2}}

　t 統計値が t 分布上である値を超えると帰無仮説を棄却します．片側検定の場合，t 分布曲線下の面積が0.05以下になる点を超えれば統計学的有意と判定します．両側検定の場合は0.025以下になる点を超えれば統計学的有意と判定します．統計学的有意となる t 統計値は自由度，つまり標本数により変化します．詳細は下記リンクを参照して下さい．

参照：t分布